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  "body": "\n## Definition\n\nLet $K$ be a field, let $N \\ge 1$, and let $\\phi \\colon \\mathbb{P}^N \\to \\mathbb{P}^N$ be a morphism of [[projective space|ZT9A6JK8QF]] defined over $K$.\nFor $N = 1$ such a morphism is given by a [[rational function|0XDK9CGYKJ]] $\\phi \\in K(z)$.\nWrite $\\phi^n$ for the $n$-fold composition of $\\phi$ with itself and $\\phi^0 = \\id$.\nFor a point $P \\in \\mathbb{P}^N(K)$:\n\n1. the *forward orbit* of $P$ is $\\mathcal{O}_\\phi(P) = \\set{\\phi^n(P) : n \\ge 0}$;\n2. $P$ is *periodic* if $\\phi^n(P) = P$ for some $n \\ge 1$;\n3. $P$ is *preperiodic* if $\\phi^m(P)$ is periodic for some $m \\ge 0$, equivalently if $\\mathcal{O}_\\phi(P)$ is finite.\n\n*Arithmetic dynamics* is the study of these orbits when $K$ is a field of arithmetic interest.\nExamples of such fields are an [[algebraic number field|MWBDFS2KJ5]], a $p$-adic field or a finite field.\n\n## Remarks\n\nThe set of preperiodic points of $\\phi$ in $\\mathbb{P}^N(K)$ is written $\\operatorname{PrePer}(\\phi, K)$.\n\nExample: let $K = \\Q$ and $\\phi(z) = z^2 - 1$.\nThen $0 \\mapsto -1 \\mapsto 0$, so $0$ and $-1$ are [[periodic|T75CZ5H47N]] of period $2$.\nAlso $1 \\mapsto 0$, so $1$ is preperiodic but not periodic, since $\\phi^n(1) \\in \\set{0, -1}$ for all $n \\ge 1$.\n\nNon-example: for the same $\\phi$, the point $2$ is not preperiodic.\nIts orbit is $2, 3, 8, 63, \\ldots$, and $\\phi(z) \u003e z \\ge 2$ for $z \\ge 2$, so the orbit is strictly increasing and infinite.\n\nWhen $K$ is a number field and $\\phi$ has degree at least $2$, Northcott's theorem on [[heights|F1FF505PWH]] implies that $\\operatorname{PrePer}(\\phi, K)$ is finite.\n",
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